Counting 682021 COCS 222 Discrete Structures 1 Basic

Counting 6/8/2021 COCS 222 - Discrete Structures 1

Basic Counting Principles Counting problems are of the following kind: “How many different 8 -letter passwords are there? ” “How many possible ways are there to pick 11 soccer players out of a 20 -player team? ” Most importantly, counting is the basis for computing probabilities of discrete events. (“What is the probability of winning the lottery? ”) 6/8/2021 COCS 222 - Discrete Structures 2

Basic Counting Principles The sum rule: If a task can be done in n 1 ways and a second task in n 2 ways, and if these two tasks cannot be done at the same time, then there are n 1 + n 2 ways to do either task. Example: The department will award a free computer to either a CS student or a CS professor. How many different choices are there, if there are 530 students and 15 professors? There are 530 + 15 = 545 choices. 6/8/2021 COCS 222 - Discrete Structures 3

Basic Counting Principles Generalized sum rule: If we have tasks T 1, T 2, …, Tm that can be done in n 1, n 2, …, nm ways, respectively, and no two of these tasks can be done at the same time, then there are n 1 + n 2 + … + nm ways to do one of these tasks. 6/8/2021 COCS 222 - Discrete Structures 4

Basic Counting Principles The product rule: Suppose that a procedure can be broken down into two successive tasks. If there are n 1 ways to do the first task and n 2 ways to do the second task after the first task has been done, then there are n 1 n 2 ways to do the procedure. 6/8/2021 COCS 222 - Discrete Structures 5

Basic Counting Principles Example: How many different license plates are there that containing exactly three English letters ? Solution: There are 26 possibilities to pick the first letter, then 26 possibilities for the second one, and 26 for the last one. So there are 26 26 26 = 17576 different license plates. 6/8/2021 COCS 222 - Discrete Structures 6

Basic Counting Principles Generalized product rule: If we have a procedure consisting of sequential tasks T 1, T 2, …, Tm that can be done in n 1, n 2, …, nm ways, respectively, then there are n 1 n 2 … nm ways to carry out the procedure. 6/8/2021 COCS 222 - Discrete Structures 7

Basic Counting Principles The sum and product rules can also be phrased in terms of set theory. Sum rule: Let A 1, A 2, …, Am be disjoint sets. Then the number of ways to choose any element from one of these sets is |A 1 A 2 … Am | = |A 1| + |A 2| + … + |Am|. Product rule: Let A 1, A 2, …, Am be finite sets. Then the number of ways to choose one element from each set in the order A 1, A 2, …, Am is |A 1 A 2 … Am | = |A 1| |A 2| … |Am|. 6/8/2021 COCS 222 - Discrete Structures 8

Inclusion-Exclusion How many bit strings of length 8 either start with a 1 or end with 00? Task 1: Construct a string of length 8 that starts with a 1. There is one way to pick the first bit (1), two ways to pick the second bit (0 or 1), two ways to pick the third bit (0 or 1), . two ways to pick the eighth bit (0 or 1). Product rule: Task 1 can be done in 1 27 = 128 ways. 6/8/2021 COCS 222 - Discrete Structures 9

Inclusion-Exclusion Task 2: Construct a string of length 8 that ends with 00. There are two ways to pick the first bit (0 or 1), two ways to pick the second bit (0 or 1), . . two ways to pick the sixth bit (0 or 1), one way to pick the seventh bit (0), and one way to pick the eighth bit (0). Product rule: Task 2 can be done in 26 = 64 ways. 6/8/2021 COCS 222 - Discrete Structures 10

Inclusion-Exclusion Since there are 128 ways to do Task 1 and 64 ways to do Task 2, does this mean that there are 192 bit strings either starting with 1 or ending with 00 ? No, because here Task 1 and Task 2 can be done at the same time. When we carry out Task 1 and create strings starting with 1, some of these strings end with 00. Therefore, we sometimes do Tasks 1 and 2 at the same time, so the sum rule does not apply. 6/8/2021 COCS 222 - Discrete Structures 11

Inclusion-Exclusion If we want to use the sum rule in such a case, we have to subtract the cases when Tasks 1 and 2 are done at the same time. How many cases are there, that is, how many strings start with 1 and end with 00? There is one way to pick the first bit (1), two ways for the second, …, sixth bit (0 or 1), one way for the seventh, eighth bit (0). Product rule: In 25 = 32 cases, Tasks 1 and 2 are carried out at the same time. 6/8/2021 COCS 222 - Discrete Structures 12

Inclusion-Exclusion Since there are 128 ways to complete Task 1 and 64 ways to complete Task 2, and in 32 of these cases Tasks 1 and 2 are completed at the same time, there are 128 + 64 – 32 = 160 ways to do either task. In set theory, this corresponds to sets A 1 and A 2 that are not disjoint. Then we have: |A 1 A 2| = |A 1| + |A 2| - |A 1 A 2| This is called the principle of inclusion-exclusion. 6/8/2021 COCS 222 - Discrete Structures 13

Tree Diagrams How many bit strings of length four do not have two consecutive 1 s? Task 1 Task 2 Task 3 Task 4 (1 st bit) (2 nd bit) (3 rd bit) 0 1 1 There are 8 strings. 6/8/2021 0 0 1 COCS 222 - Discrete Structures (4 th bit) 0 1 0 1 0 14

The Pigeonhole Principle The pigeonhole principle: If (k + 1) or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. Example 1: If there are 11 players in a soccer team that wins 12 -0, there must be at least one player in the team who scored at least twice. Example 2: If you have 6 classes from Monday to Friday, there must be at least one day on which you have at least two classes. 6/8/2021 COCS 222 - Discrete Structures 15

The Pigeonhole Principle The generalized pigeonhole principle: If N objects are placed into k boxes, then there is at least one box containing at least N/k of the objects. Example 1: In our 60 -student class, at least 12 students will get the same letter grade (A, B, C, D, or F). 6/8/2021 COCS 222 - Discrete Structures 16

The Pigeonhole Principle Example 2: Assume you have a drawer containing a random distribution of a dozen brown socks and a dozen black socks. It is dark, so how many socks do you have to pick to be sure that among them there is a matching pair? There are two types of socks, so if you pick at least 3 socks, there must be either at least two brown socks or at least two black socks. Generalized pigeonhole principle: 3/2 = 2. 6/8/2021 COCS 222 - Discrete Structures 17

Permutations and Combinations How many ways are there to pick a set of 3 people from a group of 6? There are 6 choices for the first person, 5 for the second one, and 4 for the third one, so there are 6 5 4 = 120 ways to do this. This is not the correct result! For example, picking person C, then person A, and then person E leads to the same group as first picking E, then C, and then A. However, these cases are counted separately in the above equation. 6/8/2021 COCS 222 - Discrete Structures 18

Permutations and Combinations So how can we compute how many different subsets of people can be picked (that is, we want to disregard the order of picking) ? To find out about this, we need to look at permutations. A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement of r elements of a set is called an r-permutation. 6/8/2021 COCS 222 - Discrete Structures 19

Permutations and Combinations Example: Let S = {1, 2, 3}. The arrangement 3, 1, 2 is a permutation of S. The arrangement 3, 2 is a 2 -permutation of S. The number of r-permutations of a set with n distinct elements is denoted by P(n, r). We can calculate P(n, r) with the product rule: P(n, r) = n (n – 1) (n – 2) … (n – r + 1). (n choices for the first element, (n – 1) for the second one, (n – 2) for the third one…) 6/8/2021 COCS 222 - Discrete Structures 20

Permutations and Combinations Example: P(8, 3) = 8 7 6 = 336 = (8 7 6 5 4 3 2 1)/(5 4 3 2 1) General formula: P(n, r) = n!/(n – r)! Knowing this, we can return to our initial question: How many ways are there to pick a set of 3 people from a group of 6 (disregarding the order of picking)? 6/8/2021 COCS 222 - Discrete Structures 21

Permutations and Combinations An r-combination of elements of a set is an unordered selection of r elements from the set. Thus, an r-combination is simply a subset of the set with r elements. Example: Let S = {1, 2, 3, 4}. Then {1, 3, 4} is a 3 -combination from S. The number of r-combinations of a set with n distinct elements is denoted by C(n, r). Example: C(4, 2) = 6, since, for example, the 2 combinations of a set {1, 2, 3, 4} are {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}. 6/8/2021 COCS 222 - Discrete Structures 22

Permutations and Combinations How can we calculate C(n, r)? Consider that we can obtain the r-permutation of a set in the following way: First, we form all the r-combinations of the set (there are C(n, r) such r-combinations). Then, we generate all possible orderings in each of these rcombinations (there are P(r, r) such orderings in each case). Therefore, we have: P(n, r) = C(n, r) P(r, r) 6/8/2021 COCS 222 - Discrete Structures 23

Permutations and Combinations C(n, r) = P(n, r)/P(r, r) = n!/(n – r)!/(r – r)!) = n!/(r!(n – r)!) Now we can answer our initial question: How many ways are there to pick a set of 3 people from a group of 6 (disregarding the order of picking)? C(6, 3) = 6!/(3! 3!) = 720/(6 6) = 720/36 = 20 There are 20 different ways, that is, 20 different groups to be picked. 6/8/2021 COCS 222 - Discrete Structures 24

Permutations and Combinations Corollary: Let n and r be nonnegative integers with r n. Then C(n, r) = C(n, n – r). Note that “picking a group of r people from a group of n people” is the same as “splitting a group of n people into a group of r people and another group of (n – r) people”. 6/8/2021 COCS 222 - Discrete Structures 25

Permutations and Combinations Example: A soccer club has 8 female and 7 male members. For today’s match, the coach wants to have 6 female and 5 male players on the grass. How many possible configurations are there? C(8, 6) C(7, 5) = 8!/(6! 2!) 7!/(5! 2!) = 28 21 = 588 6/8/2021 COCS 222 - Discrete Structures 26

Combinations We also saw the following: This symmetry is intuitively plausible. For example, let us consider a set containing six elements (n = 6). Picking two elements and leaving four is essentially the same as picking four elements and leaving two. In either case, our number of choices is the number of possibilities to divide the set into one set containing two elements and another set containing four elements. 6/8/2021 COCS 222 - Discrete Structures 27

Combinations Pascal’s Identity: Let n and k be positive integers with n k. Then C(n + 1, k) = C(n, k – 1) + C(n, k). How can this be explained? What is it good for? 6/8/2021 COCS 222 - Discrete Structures 28

Combinations Imagine a set S containing n elements and a set T containing (n + 1) elements, namely all elements in S plus a new element a. Calculating C(n + 1, k) is equivalent to answering the question: How many subsets of T containing k items are there? Case I: The subset contains (k – 1) elements of S plus the element a: C(n, k – 1) choices. Case II: The subset contains k elements of S and does not contain a: C(n, k) choices. Sum Rule: C(n + 1, k) = C(n, k – 1) + C(n, k). 6/8/2021 COCS 222 - Discrete Structures 29

Pascal’s Triangle In Pascal’s triangle, each number is the sum of the numbers to its upper left and upper right: 1 1 1 … 6/8/2021 2 3 4 … 1 1 3 6 … 1 4 … 1 … COCS 222 - Discrete Structures … 30

Pascal’s Triangle Since we have C(n + 1, k) = C(n, k – 1) + C(n, k) and C(0, 0) = 1, we can use Pascal’s triangle to simplify the computation of C(n, k): k C(0, 0) = 1 C(1, 1) = 1 n C(2, 0) = 1 C(2, 1) = 2 C(2, 2) = 1 C(3, 0) = 1 C(3, 1) = 3 C(3, 2) = 3 C(3, 3) = 1 C(4, 0) = 1 C(4, 1) = 4 C(4, 2) = 6 C(4, 3) = 4 C(4, 4) = 1 6/8/2021 COCS 222 - Discrete Structures 31

Binomial Coefficients Expressions of the form C(n, k) are also called binomial coefficients. How come? A binomial expression is the sum of two terms, such as (a + b). Now consider (a + b)2 = (a + b). When expanding such expressions, we have to form all possible products of a term in the first factor and a term in the second factor: (a + b)2 = a·a + a·b + b·a + b·b Then we can sum identical terms: (a + b)2 = a 2 + 2 ab + b 2 6/8/2021 COCS 222 - Discrete Structures 32

Binomial Coefficients For (a + b)3 = (a + b)(a + b) we have (a + b)3 = aaa + aab + aba + abb + baa + bab + bba + bbb (a + b)3 = a 3 + 3 a 2 b + 3 ab 2 + b 3 There is only one term a 3, because there is only one possibility to form it: Choose a from all three factors: C(3, 3) = 1. There is three times the term a 2 b, because there are three possibilities to choose a from two out of the three factors: C(3, 2) = 3. Similarly, there is three times the term ab 2 (C(3, 1) = 3) and once the term b 3 (C(3, 0) = 1). 6/8/2021 COCS 222 - Discrete Structures 33

Binomial Coefficients This leads us to the following formula: (Binomial Theorem) With the help of Pascal’s triangle, this formula can considerably simplify the process of expanding powers of binomial expressions. For example, the fifth row of Pascal’s triangle (1 – 4 – 6 – 4 – 1) helps us to compute (a + b)4: (a + b)4 = a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + b 4 6/8/2021 COCS 222 - Discrete Structures 34